[Paper Review] On the Limitations of Physics-informed Deep Learning: Illustrations Using First Order Hyperbolic Conservation Law-based Traffic Flow Models

Archie J. Huang, Shaurya Agarwal|arXiv (Cornell University)|Feb 23, 2023

Model Reduction and Neural Networks4 citations

TL;DR

This paper investigates the limitations of physics-informed deep learning (PIDL) in reconstructing first-order hyperbolic conservation laws, using the Lighthill-Whitham-Richards (LWR) traffic flow model as a case study. It demonstrates that PIDL fails to accurately capture shockwaves in the hyperbolic LWR PDE due to non-smooth solutions, whereas adding a diffusion term (parabolic variant) enables stable, accurate reconstruction through improved gradient optimization and smoothness.

ABSTRACT

Since its introduction in 2017, physics-informed deep learning (PIDL) has garnered growing popularity in understanding the evolution of systems governed by physical laws in terms of partial differential equations (PDEs). However, empirical evidence points to the limitations of PIDL for learning certain types of PDEs. In this paper, we (a) present the challenges in training PIDL architecture, (b) contrast the performance of PIDL architecture in learning a first order scalar hyperbolic conservation law and its parabolic counterpart, (c) investigate the effect of training data sampling, which corresponds to various sensing scenarios in traffic networks, (d) comment on the implications of PIDL limitations for traffic flow estimation and prediction in practice. Detailed in the case study, we present the contradistinction in PIDL results between learning the traffic flow model (LWR PDE) and its variation with diffusion. The outcome indicates that PIDL experiences significant challenges in learning the hyperbolic LWR equation due to the non-smoothness of its solution. On the other hand, the architecture with parabolic PDE, augmented with the diffusion term, leads to the successful reassembly of the density data even with the shockwaves present.

Motivation & Objective

• To investigate the challenges of training physics-informed deep learning (PIDL) models on first-order hyperbolic conservation laws in traffic flow.
• To compare PIDL performance between the hyperbolic LWR PDE and its parabolic counterpart with a diffusion term.
• To analyze the impact of training data sampling strategies on PIDL accuracy in shockwave-prone regions.
• To evaluate the practical implications of PIDL limitations for real-world traffic state estimation (TSE).
• To explore strategies such as Lagrangian sensing and increased sampling near shocks to improve PIDL performance.

Proposed method

• The study employs a physics-informed neural network (PINN) architecture trained on initial and boundary conditions of the LWR PDE and its parabolic variant.
• The hyperbolic LWR PDE is used as a baseline for modeling discontinuous traffic shockwaves, while the parabolic form includes a diffusion coefficient ε to smooth solutions.
• Training data sampling is varied across Eulerian (fixed sensors) and Lagrangian (moving vehicles) observation types to simulate real-world sensing scenarios.
• The model is evaluated on synthetic and real-world NGSIM traffic data to assess reconstruction accuracy of traffic density.
• Gradient optimization is analyzed to understand instability in hyperbolic cases due to non-smooth solutions.
• Sensitivity to the diffusion parameter ε is tested to assess its role in stabilizing training and improving reconstruction.

Figure 2: Architecture of a Physics-informed Deep Learning (PIDL) Neural Network

Experimental results

Research questions

• RQ1 Why does physics-informed deep learning fail to reconstruct the hyperbolic LWR PDE despite incorporating physical laws?
• RQ2 How does adding a diffusion term transform the PDE from hyperbolic to parabolic, and what impact does this have on PIDL training stability?
• RQ3 How do different data sampling strategies—Eulerian and Lagrangian—affect PIDL performance in regions with shockwaves?
• RQ4 Can PIDL accurately reconstruct real traffic data with discontinuities when relying only on initial and boundary conditions?
• RQ5 What role does the diffusion coefficient ε play in improving gradient optimization and solution accuracy in PIDL for conservation law-based models?

Key findings

• PIDL fails to reconstruct the hyperbolic LWR PDE due to non-smooth, discontinuous solutions that destabilize gradient optimization in the neural network.
• The addition of a diffusion term transforms the PDE into a parabolic form, enabling stable training and accurate reconstruction of traffic density, even with shockwaves present.
• The parabolic variant achieves high-precision reconstruction with minimal error, while the hyperbolic version produces significant inaccuracies near discontinuities.
• Increased sampling density along shockwave trajectories improves PIDL performance, but is insufficient to overcome fundamental instability in the hyperbolic case.
• The diffusion coefficient ε must be tuned to traffic behavior; its inclusion is essential for realistic modeling and stable learning in PIDL.
• Lagrangian sensing data and domain decomposition techniques can enhance reconstruction reliability, especially in complex, multi-scale traffic scenarios.

Figure 3: Ring Road Configuration of the Datasets